The present invention relates to a method and a device for analog-to-digital conversion of a signal.
Analog-to-digital conversion, referred to in the following as A/D conversion, devices or circuitry are used for digital signal processing, e.g., in television, radio or receiver technology, as so-called A/D converters for video and audio signals. In this context, analog signals are converted into digital signals for processing. U.S. Pat. No. 5,568,142, for example, purports to describe a method for analog-to-digital conversion of a band-limited signal, where, on the basis of filters, the signal is partitioned into various signals which, once digitized, are recombined by filters.
The constant rise in memory chip capacity and the increasing power of high-speed processors have resulted in improved performance of digital signal processing. With respect to resolution and bandwidth, the performance of A/D converters is improving at a substantially slower rate than that of comparable components used in digital signal processing. The performance of A/D converters is limited by a constant product of resolution and bandwidth (see R. H. Walden, xe2x80x9cPerformance Trends for ADCxe2x80x9d, IEEE Communication Magazine, February 1999, pp. 96-101). Therefore, to enhance performance, in particular to achieve a highest possible bandwidth, a plurality of conventional A/D converters having time interleaved sampling instants is employed. The disadvantage here is that offset and gain errors resulting from the parallel configuration of the A/D converters cause jump discontinuities to occur at the sampling instants. These jump discontinuities are characterized by discrete disturbance lines, or lines of perturbation, in the useful signal spectrum.
An object of the present invention is, therefore, to provide a method and a device for the analog-to-digital conversion of an analog signal, which will render possible high performance with respect to bandwidth and resolution.
The present invention provides a method for the analog-to-digital conversion of a band-limited signal (x(t)), wherein the signal (x(t)) is transformed on the basis of orthogonal functions (gj(t)), coefficients (aj) corresponding to the orthogonal functions (gj(t)) and the signal x(t) being defined and digitized, and, on the basis of the digitized coefficients (ajd), the signal (xd(t)) being inversely transformed in the digital domain by orthogonal functions (hj(t)).
In this context, instead of sequentially digitizing individual sampled values of a conventional A/D converter, the method according to the present invention processes one complete interval of the signal""s time function. To this end, the signal that is time-limited to the interval is described on the basis of orthogonal functions. The signal is preferably decomposed into several intervals. By limiting the time function of the signal to the interval, with subsequent transformation with the assistance of orthogonal functions, the signal is substantially fully defined in the digital domain on the basis of discrete coefficients of the orthogonal functions in equidistant or non-equidistant spacing, and can be reconstructed from these coefficients. In other words: on the basis of orthogonal functions, the signal is processed into an equation for its transforms, which is then digitized and inversely transformed into the original domain, with the result that the original function of the signal is defined in the digital domain.
The signal is expediently limited in the time domain to the interval and, within the interval, represented by a sum of orthogonal functions having a predefinable number of summands, or addends, the coefficients corresponding to the orthogonal functions being defined for the interval and digitized, and, through inverse transformation of the digitized coefficients on the basis of orthogonal functions, the signal being represented in the digital domain. The signal is preferably decomposed into several intervals, enabling it to be represented over a large time domain. In band-limiting the signal, it is useful to consider the sampling, or Nyquist, theorems. According to the sampling theorems, when limiting the time or frequency function, discrete values of the frequency function or of the time function suffice for providing a complete description of the signal. The time function of the signal is preferably represented by the development of orthogonal functions in accordance with a complete system. This means that the band-limited signal is fully described by a finite summation. For example, the signal in the analog domain is represented on the basis of the generalized Fourier analysis:                                           x            ⁡                          (              t              )                                =                                                    ∑                j                N                            ⁢                                                a                  j                                ·                                                      g                    j                                    ⁡                                      (                    t                    )                                                                        =                                          ∑                j                N                            ⁢                                                (                                                            x                      ⁡                                              (                        t                        )                                                              ,                                                                  g                        j                                            ⁡                                              (                        t                        )                                                                              )                                ·                                                      g                    j                                    ⁡                                      (                    t                    )                                                                                      ,                            (        1        )                                          a          j                =                              (                                          x                ⁡                                  (                  t                  )                                            ,                                                g                  j                                ⁡                                  (                  t                  )                                                      )                    =                                    ∫              0              T                        ⁢                                                            x                  ⁡                                      (                    t                    )                                                  ·                                                      g                    j                                    ⁡                                      (                    t                    )                                                              ⁢                              ⅆ                t                                                                        (        2        )            
where x(t)=time function of the signal, gj(t)=orthogonal functions, aj=coefficients, N=number of summands=number of orthogonal functions=number of interpolation nodes in the transformed domain (frequency domain for the special case of the Fourier transform)=number of parallel channels, T=length of the interval in the time domain.
Equation (2) defines the so-called inner product between x(t) and gj(t). For the sake of brevity, the symbolic notation (x(t), gj(t)) is used in the following text. The closeness of the approximation is determined by the number of summands, which, in a real system, are truncated following a finite number. In this context, the minimal value for the number N of summands, also referred to as interpolation nodes, is derived from the sampling theorems in the time and frequency domain for time-limited and band-limited signals. The number of summands N is preferably determined by the equation:                     N        =                  T          τ                                    (        3        )            
where T=the length of the interval in the time domain,
xcfx84=the segment in the time domain,
xe2x80x83where                     τ        =                              1                          2              ⁢              B                                ⁢                      xe2x80x83                    ⁢                      (                          Nyquist              ⁢                              xe2x80x83                            ⁢              criterion                        )                                              (        4        )            
and B=bandwidth.
In this context, the number of summands is preferably selected such that adequate resolution is assured. Preferably, one selects the same systems of orthogonal functions in the analog domain (transform) and in the digital domain (inverse transform). Alternatively, the systems of orthogonal functions, also referred to as basic functions, may also be different.
The digitized coefficients are expediently inversely transformed in such a way that, in the digital domain, the signal is described by multiplying the digitized coefficients by predefinable orthogonal functions, and through subsequent summation. In the case that the basic functions differ in the analog and in the digital domain, then the coefficients are combined by a linear transform, as expressed by:                                           x            ⁡                          (              t              )                                =                                                    ∑                j                N                            ⁢                                                                    a                    j                                    ·                                      g                    j                                                  ⁢                                  (                  t                  )                                                      =                                          ∑                j                N                            ⁢                                                b                  j                  d                                ·                                                      h                    j                                    ⁡                                      (                    t                    )                                                                                      ,                            (        5        )            
provided that gj(t)xe2x89xa0hj(t),
xe2x80x83respectively                                           x            ⁡                          (              t              )                                =                                                    ∑                j                N                            ⁢                                                                    a                    j                                    ·                                      g                    j                                                  ⁢                                  (                  t                  )                                                      =                                          ∑                j                N                            ⁢                                                a                  j                  d                                ·                                                      h                    j                                    ⁡                                      (                    t                    )                                                                                      ,                            (        6        )            
provided that gj(t)=hj(t),
where x(t)=time function of the signal, gj(t)=orthogonal functions in the analog domain, aj=coefficients in the analog domain, hj(t)=orthogonal functions in the digital domain, ajd, bjd=coefficients in the digital domain, N=number of summands.
Depending on the requirements and criteria for the digital signal processing, trigonometric functions, Walsh functions, and/or complex exponential functions are used as orthogonal functions. In the analog domain, trigonometric functions, e.g., sine functions and/or cosine functions, are preferably used. In the digital domain, functions, such as Walsh or Haar functions, which can only assume the values +1 or xe2x88x921, may be employed.
In an orthonormal system, it holds for the inner product of orthogonal functions that:                     g        j            ,              g        i              )    =            {                                                                                    0                  ,                                                                                                  when                    ⁢                                          xe2x80x83                                        ⁢                    j                                    ≠                  i                                                                                                      1                  ,                                                                                                                        when                      ⁢                                              xe2x80x83                                            ⁢                      j                                        =                    i                                    ;                                                              ⁢                      xe2x80x83                    ⁢                      h            j                          ,                  h          i                    )        =          {                                                  0              ,                                                                          when                ⁢                                  xe2x80x83                                ⁢                j                            ≠              i                                                                          1              ,                                                                                            when                  ⁢                                      xe2x80x83                                    ⁢                  j                                =                i                            ;                                          
To determine coefficients bj in equation (5), the scalar product (inner product) is formed.                                           ∑            j                    ⁢                                    (                              x                ,                                  g                  j                                            )                        ⁢                          g              j                                      =                                            ∑              j                        ⁢                                          (                                  x                  ,                                      h                    j                                                  )                            ⁢                              h                j                                              ⁢                      xe2x80x83                    |                                    h              i                        ⁢                          xe2x80x83                        ⁢            formation            ⁢                          xe2x80x83                        ⁢            of            ⁢                          xe2x80x83                        ⁢            the            ⁢                          xe2x80x83                        ⁢            inner            ⁢                          xe2x80x83                        ⁢            product                                              (        7        )                                                      ∑            j                    ⁢                                    (                              x                ,                                  g                  j                                            )                        ⁢                          xe2x80x83                        ⁢                          (                                                g                  j                                ,                                  h                  i                                            )                                      =                  (                      x            ,                          h              i                                )                                    (        8        )                                                      ∑            j                    ⁢                                    a              j                        ⁡                          (                                                g                  j                                ,                                  h                  i                                            )                                      =                  b          i                                    (        9        )            
In this context, the coefficients in the digital domain are preferably determined on the basis of a transformation matrix having matrix elements (gj, hi)=mj,i as expressed by:                               (                                                    ⋮                                                                                      b                  i                                                                                    ⋮                                              )                =                              (                                                            ⋯                                                  ⋯                                                                              ⋮                                                  ⋮                                                                              ⋯                                                                      (                                                                  g                        j                                            ,                                              h                        i                                                              )                                                                        )                    ⁢                      xe2x80x83                    ⁢                      (                                                            ⋯                                                                      a                    j                                                                    ⋯                                                      )                                              (        10        )            
In an embodiment according to the present invention, in place of the above described decomposition (representation) of the signal function according to a system of classical orthogonal functions (generalized Fourier transform), a decomposition (representation) is undertaken through a wavelet transformation.
The wavelet transformation for the analog signal x(t) is defined as follows:                                           L            ψ                    ⁢                      x            ⁡                          (                              a                ,                b                            )                                      =                              1                                          "LeftBracketingBar"                a                "RightBracketingBar"                                              ·                                    ∫                              -                ∞                            ∞                        ⁢                                                            x                  ⁡                                      (                    t                    )                                                  ·                                  ψ                  ⁡                                      (                                                                  t                        -                        b                                            a                                        )                                                              ⁢                              ⅆ                t                                                                        (        11        )            
where Lxcexa8x (a,b)=wavelet transformation of the signal with two variables, a=extension on the time axis, b=shift on the time axis, x(t)=time function of the signal, xcexa8(txe2x88x92b/a)=vavelet function.
The inverse transformation of the signal is preferably defined as follows:                               x          ⁡                      (            t            )                          =                  c          ·                                    ∫                              -                ∞                            ∞                        ⁢                                          ∫                                  -                  ∞                                ∞                            ⁢                                                L                  ψ                                ⁢                                                      x                    ⁡                                          (                                              a                        ,                        b                                            )                                                        ·                                                            Ψ                                              a                        ,                        b                                                              ⁡                                          (                      t                      )                                                        ·                                      1                                          a                      2                                                                      ⁢                                  xe2x80x83                                ⁢                                  ⅆ                  a                                ⁢                                  xe2x80x83                                ⁢                                  ⅆ                  b                                                                                        (        12        )            
where Lxcexa8x (a,b)=wavelet transformation of the signal with two variables, a=extension on the time axis, b=shift on the time axis, x(t)=time function of the signal, xcexa8(t)=wavelet function, c=constant.
The double integral is realized redundantly. A double sum may also be alternatively employed.
It appears to be especially advantageous for the application intended here to use the Haar-Wavelet function, as expressed by:                               Ψ          ⁡                      (            t            )                          =                  {                                                    1                                                                                  for                    ⁢                                          xe2x80x83                                        ⁢                    0                                    ≤                  t                  ≤                                      1                    /                    2                                                                                                                        -                  1                                                                                                  for                    ⁢                                          xe2x80x83                                        ⁢                                          1                      /                      2                                                        ≤                  t                  ≤                  1                                                                                    0                                                              otherwise                  .                                                                                        (        13        )            
Thus, the wavelet transformation is derived as expressed by equation (11):                                           L            ψ                    ⁢                      x            ⁡                          (                              a                ,                b                            )                                      =                              1                                          "LeftBracketingBar"                a                "RightBracketingBar"                                              ·                      (                                                            ∫                  b                                      b                    +                                          a                      2                                                                      ⁢                                                      x                    ⁡                                          (                      t                      )                                                        ⁢                                      ⅆ                    t                                                              -                                                ∫                                      b                    +                                          a                      2                                                                            b                    +                    a                                                  ⁢                                                      x                    ⁡                                          (                      t                      )                                                        ⁢                                      ⅆ                    t                                                                        )                                              (        14        )            
Another preferred embodiment of the wavelet transformation is the dyadic wavelet transformation, in which wavelet functions are used as basic functions, as expressed by the following relation:                                           Ψ                          i              ,              j                                ⁡                      (            t            )                          =                                            1                                                2                  i                                                      ·            Ψ                    ⁢                      xe2x80x83                    ⁢                      (                                          t                -                                                      2                    i                                    ·                  j                                                            2                i                                      )                                              (        15        )            
The various basis functions for the dyadic wavelet transformation are derived from a wavelet function by doubling or halving the width and by shifting the width by integral multiples. The time intervals and, respectively, instants a, b for the wavelet functions are advantageously generated through frequency division, i.e., deceleration of a fast basic clock rate. This can be accomplished, for example, by shift registers.
The present invention also provides a device for analog-to-digital conversion of an analog signal having an input model for transforming the signal in the analog domain using orthogonal functions, a module for digitizing coefficients of the transformation function, as well as an output module for inversely transforming the signal in the digital domain. The input module is effectively used to represent the signal within the interval by a sum of orthogonal functions having a predefinable number of summands. The entire signal is expediently decomposed into a plurality of intervals. The coefficients are preferably defined for the interval by the input module.
After forming the inner products (also referred to as scalar products), i.e., after determining the coefficients within the interval that correspond to the orthogonal functions, the N coefficients are digitized in N modules, in particular in N conventional A/D converters. By multiplying the coefficients determined in the process by the orthogonal functions using the output module, the signal is completely representable in the digital domain.
In one embodiment according to the present invention, the input module contains a number of correlators equivalent to the number of summands, each correlator encompassing one multiplier and integrator. In this context, the multiplier in question is used to multiply the time function of the signal by the corresponding orthogonal function. Expediently provided as an integrator is a low-pass filter. The combination, in terms of circuit engineering, of the multiplier and low-pass filter enables coefficients aj of the orthogonal function to be determined in the analog domain.
The input module optionally contains, by preference, a number of matched filters equivalent to the number of summands. Each so-called matched filter, also referred to as signal-adapted filter or correlation filter, preferably encompasses a filter having an impulse response representing the particular orthogonal function, and a sampler. Thus, in terms of circuit engineering, an especially simple design of the signal-transformation device is provided.
Once the inner products are formed, with the result that the coefficients of the orthogonal functions are determined in the analog domain, the coefficients are digitized by the conventional A/D converter. The number of correlators, i.e., of correlation filters (matched filters) is preferably equivalent to the number of A/D converters. In the digital domain, the output module usefully contains a number of multipliers equivalent to the number of summands, as well as a summation element.
The multiplier is used to multiply the coefficient in question by the orthogonal function in the digital domain. Subsequent summation of all parallel branches, or subtrees, makes the signal completely representable in the digital domain. The number of branches or channels is equivalent in this context to the number of summands. Each branch includes, on the input side, the correlator or correlation filter for transformation, the corresponding A/D converter for digitization, and, on the output side, the multiplier for inverse transformation. A device featuring this kind of circuit-engineering design and used for analog-to-digital signal conversion is also characterized as a correlation analog-to-digital converter.
An embodiment of the device according to the present invention uses dyadic Haar wavelets as orthogonal functions. Each input module includes two parallel-connected switches, each having an assigned low-pass filter. In this context, the two switches of each branch are used to form the integral over signal x(t), with the limits of integration conforming to the selected wavelet function. Each branch represents a different wavelet function, or basic function, the various wavelet functions being generated from a logic function (see equation (13)) by doubling or halving the width and by shifting the width by integral multiples. These operations are performed by the switches. To this end, the opening and closing instants for the switches are selected to achieve an integration in conformance with equation (14).
The coefficients of the wavelet transformation are preferably digitized for each branch by the corresponding module, in particular by a conventional A/D converter. To normalize the discrete coefficients as a function of the corresponding wavelet function, a matching element, or adaptor, is expediently provided for each branch. The normalization is preferably carried out in the digital domain. For the inverse transformation, the output module advantageously contains a number of matching elements and multipliers equivalent to the number of branches. Subsequent summation of all branches makes the signal thoroughly representable in the digital domain. This means that, in the case of the dyadic Haar wavelet transformation, each branch includes, on the input side, the two switches, each having a corresponding low-pass filter, for the transformation, the corresponding A/D converter for the digitization, and, on the output side, the matching element and the multiplier for the inverse transformation.
It is beneficial for a receiver to include one of the above described devices for the analog-to-digital conversion of a signal. This makes the receiver particularly suited for digital audio broadcasting, where high-speed signal processing with an especially high sampling rate is required.
According to the present invention, a combined time-domain and frequency-domain analysis is performed on the signal. By considering the signal in a time interval, as well as by describing it on the basis of orthogonal functions within the interval, the signal is completely described, both in the analog as well as in the digital domain, which, in contrast to sampling operations using conventional A/D converters (time-interleaving A/D converters), makes it possible to reliably avoid offset and gain errors and resultant discrete disturbance, or perturbation, lines. In comparison to a single, conventional A/D converter having a high sampling rate, the multiplicity of parallel-connected A/D converters, with the number of parallel branches being equivalent to the number of predefinable summands, means that the sampling rate of the single A/D converter of the device should be selected to be lower by the factor of the number of summands.